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多項式方程組的求解是數學中的經典問題。今天，多項式模型無處不在，並在科學中廣泛使用，如機器人技術、編碼理論、優化、數學生物學、電腦視覺、博弈論、統計學及許多其他領域。《多項式方程組的求解（英文版）/美國數學會經典影印系列》提供了跨越數學學科的橋樑，揭示了多項式方程組的許多方面。它涵蓋了廣泛的數學技巧和演算法，包括符號計算和數值計算。
多項式方程組的解集是代數變數——代數幾何的基本物件。代數變數的演算法研究是計算代數幾何的核心主題。幾何計算軟體的新發展令人興奮，已經徹底改變了這個領域。以前棘手的問題已易於處理，這為實驗和猜想提供了沃土。
《多項式方程組的求解（英文版）/美國數學會經典影印系列》的前半部分簡要介紹了計算代數幾何的新技術，即代數簇的演算法研究；後半部分從各種新穎和意想不到的角度探討了多項式方程，介紹了學科間的聯繫，討論了當前研究的重點，並概述了未來可能的演算法。
整本書中有許多動手實例和練習，包括Mapie，MATLAB，Macaulay 2，Singular，PH Cpack，SOS Tools和CoCoA的簡短但完整的會話。這些例子對沒有代數幾何或交換代數背景的讀者特別有用。幾分鐘之內，讀者就能學會如何輸入多項式方程，並在電腦螢幕上看到一些有意義的結果。
讀者需要具備基本的抽象和計算代數知識。本書適合作為計算代數方向的研究生課程教材。
Chapter 1．Polynomials in One Variable
1．1．The Fundamental Theorem of Algebra
1．2．Numerical Root Finding
1．3．Real Roots
1．4．Puiseux Series
1．5．Hypergeometric Series
1．6．Exercises
Chapter 2．GrSbner Bases of Zero-Dimensional Ideals
2．1．Computing Standard Monomials and the Radical
2．2．Localizing and Removing Known Zeros
2．3．Companion Matrices
2．4．The Trace Form
2．5．Solving Polynomial Equations in Singular
2．6．Exercises
Chapter 3．Bernstein's Theorem and Fewnomials
3．1．From Bzout's Theorem to Bernstein's Theorem
3．2．Zero-dimensional Binomial Systems
3．3．Introducing a Toric Deformation
3．4．Mixed Subdivisions of Newton Polytopes
3．5．Khovanskii's Theorem on Fewnomials
3．6．Exercises
Chapter 4．Resultants
4．1．The Univariate Resultant
4．2．The Classical Multivariate Resultant
4．3．The Sparse Resultant
4．4．The Unmixed Sparse Resultant
4．5．The Resultant of Four Trilinear Equations
4．6．Exercises
Chapter 5．Primary Decomposition
5．1．Prime Ideals， Radical Ideals and Primary Ideals
5．2．How to Decompose a Polynomial System
5．3．Adjacent Minors
5．4．Permanental Ideals
5．5．Exercises
Chapter 6．Polynomial Systems in Economics
6．1．Three-Person Games with Two Pure Strategies
6．2．Two Numerical Examples Involving Square Roots
6．3．Equations Defining Nash Equilibria
6．4．The Mixed Volume of a Product of Simplices
6．5．Computing Nash Equilibria with PHCpack
6．6．Exercises
Chapter 7．Sums of Squares
7．1．Positive Semidefinite Matrices
7．2．Zero-dimensional Ideals and SOStools
7．3．Global Optimization
7．4．The Real Nullstellensatz
7．5．Symmetric Matrices with Double Eigenvalues
7．6．Exercises
Chapter 8．Polynomial Systems in Statistics
8．1．Conditional Independence
8．2．Graphical Models
8．3．Random Walks on the Integer Lattice
8．4．Maximum Likelihood Equations
8．5．Exercises
Chapter 9．Tropical Algebraic Geometry
9．1．Tropical Geometry in the Plane
9．2．Amoebas and their Tentacles
9．3．The Bergman Complex of a Linear Space
9．4．The Tropical Variety of an Ideal
9．5．Exercises
Chapter 10．Linear Partial Differential Equations with Constant Coefficients
10．1．Why Differential Equations?
10．2．Zero-dimensional Ideals
10．3．Computing Polynomial Solutions
10．4．How to Solve Monomial Equations
10．5．The Ehrenpreis-Palamodov Theorem
10．6．Noetherian Operators
10．7．Exercises
Bibliography
Index
